(otherwise said the manifold is of positive type) the solution
of (REF ) is not unique, while uniqueness holds, up to symmetries,
if \(Q(M)\le 0\) . At this point people start asking wheter if, given
a manifold of positive type, the family of solutions of (REF )
is \(C^{2}\)  compact, that is there is a \(C^{2}\)  uniform estimate
on the set of solutions. The question has been solved by S. Brendle,
M. A. Khuri, F. C. Marques and R. Schoen in a series of works [1]}, [2]}, [3]}.
Their result can be summarized as follows: compactness holds for dimensions
\(3\le n\le 24\)  if the manifold is not conformally equivalent to a
round sphere, while for \(n\ge 25\)  it is possible to construct manifolds
for which the set of solutions is not a compact set.
